Limitations and tradeoffs in synchronization of large-scale networks with uncertain links

TL;DR

This paper analyzes how uncertainty in links affects synchronization in large-scale nonlinear networks, providing conditions, tradeoffs, and optimal parameters for achieving synchronization.

Contribution

It offers an analytical framework for understanding the interplay of dynamics, topology, and uncertainty in network synchronization, including optimal connection strategies.

Findings

Synchronization condition independent of network size

Existence of an optimal number of neighbors for maximum synchronization margin

Analytical formula for optimal gain in network synchronization

Abstract

We study synchronization in scalar nonlinear systems connected over a linear network with stochastic uncertainty in their interactions. We provide a sufficient condition for the synchronization of such network systems expressed in terms of the parameters of the nonlinear scalar dynamics, the second and largest eigenvalues of the mean interconnection Laplacian, and the variance of the stochastic uncertainty. The sufficient condition is independent of network size thereby making it attractive for verification of synchronization in a large size network. The main contribution of this paper is to provide analytical characterization for the interplay of roles played by the internal dynamics of the nonlinear system, network topology, and uncertainty statistics in network synchronization. We show there exist important tradeoffs between these various network parameters necessary to achieve synchronization. We show for nearest neighbor networks with stochastic uncertainty in interactions there exists an optimal number of neighbors with maximum margin for synchronization. This proves in the presence of interaction uncertainty, too many connections among network components is just as harmful for synchronization as the lack of connection. We provide an analytical formula for the optimal gain required to achieve maximum synchronization margin thereby allowing us to compare various complex network topology for their synchronization property.